The \(\Gamma \) -limit of such a functional as \(\beta \rightarrow +\infty \)  (namely with local attractive term) has been characterized in [1]}.
In suitable regimes, minimizers of the limit functionals have been proved to be given by periodic unions of stripes (i.e., intervals in one dimension, see [1]}) with techniques developed in [3]}, [4]}, [1]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. For characterization of minimizers with power law attractive-repulsive potentials see [12]}.
